Theta-bi Function
The Theta-bi Function, written mathematically as \theta''mb→b '', is a function for the limit of x'', where ''x is the smallest strongly inconsistent ordinal that can be put into a huge cardinal axiom to turn it into an n''-meaningless cardinal. Apart from being a mathematical curiosity, the Theta-bi Function is also very useful for compressing set-theoretic files of the form 'M'(''x). The simplicity of compressing M''(x) files using the Theta-bi Function led to ''M(x'') files becoming the most widely used file format. Discovery The Theta-bi Function was discovered by the mathematician and computer scientist Alex Yornunln. He immediately realized its possible usefulness, and potential to eliminate the crude technique for information sending of compressing files using the elementary subset duality, which took huge amounts of energy, magic, and brute-force computing. He thought that his function could be used to compress 'M'(''x) files quickly and easily. He tested it, trying to compress a file of the form {M''(x''(''x(x''(''x(x''(''x(x''(''x)))))))) → ''M(x''(''x(x''))), where ''x = M''(x) \mu 2 + \epsilon3}. He was able to do it on his own computer in several hours, which was an astonishing feat. Any ''M(x'') file which used \mu 2 + \epsilon3 had always been particularly hard to calculate, since it is larger than any elementary subset by definition. This was a breakthrough. The Theta-bi Function quickly became the most popular method of file compression for information sending. Mathematical Principles The Theta-bi Functon is the limit of \mu n + \epsilonn, which is the in-compressible axiom for huge and superhuge cardinals. It therefore greatly exceeds the compressible axiom, and represents ''x, where x'' is the smallest strongly inconsistent ordinal that can be put into a huge cardinal axiom to turn it into an ''n-meaningless cardinal. The Theta-bi Function, being the limit of x'', is the limit of ''n-meaningless cardinals in general. In fact, all n''-meaningless cardinals can be modeled using the Theta-bi Function. For example, M((X)X2) can be expressed using the Theta-bi Function as \thetaM(2)b→bx, an extraordinarily compact expression of an X2*2-meaningless cardinal. The Theta-bi Function is created from \theta , the determining variable for lower level metafractal constructions, ''m, the limit of huge cardinals, and b'', a high-order inconsistent variable in superhuge cardinal theory. The exact mathematical formulation of ''b is {M(x)→''x''}→{[M(x]←b}M(b). The proof that mb→b; that is, that mb ''can be embedded into ''b is on the order of 101010102.7 pages long. Combining mb→b with \theta , with mb→b as a subscript (meaning that it is the mb→b''th function of \theta ), means iteratively applying the ''concept of mb being embeddable into b'' to the \theta metafractal construction. This results in a ''very high order function. Perhaps the most curious and useful property of the Theta-bi Function is that it can be either consistent, or inconsistent depending on how it is expressed. It was at first thought that this was some sort of meta-inconsistency, but it was proven that consistence and inconsistence are conceptually too high to be rolled up into a meta-concept, when applied to something such as the Theta-bi Function. This quirk of the Theta-bi Function is very odd, since it is typically only found in unprovable duel expressions and related branches of mathematics. The reason that the Theta-bi Function is consistent is very simple. It is a well known fact that the limit of any inconsistent mathematical construct (and x'' is an inconsistent mathematical construct) must be consistent. Mathematically, this is proven like so: {''C= lim. I'' {''I→''I'' {I'' = ''Xn(I'')}}}: {lim. ''I = I''( \xi )}, {[1. ''I( \xi ) is expressible as Xn(I''( \xi ))]}, {''Xn(I''( \xi )) > ''Xn(I'') {''Xn < X''( \xi ), ''I < I''( \xi )}} [2. ''X( \xi ) and I''( \xi ) = unachievable, ''Xn(I''( \xi ))→''Xn(I'') = consistent. Ignore achievability in following calculation within | symbols (basis: achievability = high-order trivial concept (unachievable things can be conceptualized (eg. thought of)and plugged into thought constructs "by hand", and can be theoretically accomplished))]: |''Xn(I''( \xi )) = ''Xn(I'')| [3. Since ''I = Xn(I'') (''I represents any inconsistent mathematical construct (Def. of inconsistency = def. in transfinite and higher-order mathematics)), and (ignoring achievability for reasons stated in (2.)) Xn(I'') = ''Xn(I''( \xi )), it follows logically that ''I = Xn(I''( \xi )). This means that under high-order triviality (It was stated and proven in (2.) that achievability is a concept that is of a high-order trivial nature) it is not possible to complete the embedding of ''I into itself (where embedding I'' into itself = ''I→''I'') if I'' = ''I( \xi ) because the complete calculation of I''( \xi ) is unachievable, and complete calculation of the constant into which another constant is being embedded, and of the constant that is being embedded into another constant, is necessary for embedding (as can be proven mathematically). If ''I→''I'' (where I'' = ''I( \xi )) cannot be completely embedded into itself under high-order triviality, that means that I'' is consistent under high-order triviality (def. of consistency = def. in transfinite and higher-order mathematics.). Since ''I (where I'' = ''I( \xi )) = lim. I'', lim. ''I = consistent. If C'' = lim. ''I, then};--''C = consistent. This is a very simple proof that the the limit of any inconsistent function is consistent. Just substitute ''C for \theta''mb→b '' and I'' for ''x, and the proof will show that the Theta-bi Function is consistent. Expressing it as inconsistent is much more difficult. The first step is to map the nn \omega function to the value of the Shaurlot system. This alone requires approximately 10106 pages to accomplish. Once it is accomplished, the resulting function can generate huge and superhuge cardinals.